It is interesting to see how Pythagoras Theorem helps in identifying the location of an irrational number on the Number Line. Consider a number x which is a rational number but not a perfect square. It follows that the square root of x must be irrational, that is, a non-terminating and non-recurring decimal number. Now, our interest is to determine where this lies on the Number Line. To do this, let us consider a right triangle whose base equals (x-1)/2 the hypotenuse equals (x+1)/2. What would be the height of this right triangle, we mean, the other arm of the right triangle? Let us suppose it is y. Pythagoras Theorem tells us that the sum of squares of the arms of a right triangle equals the square of the hypotenuse. So, in the triangle we have just considered, we can write:
[(x – 1)/2]^2 + y^2 = [(x + 1)/2]^2
Or y^2 = [(x + 1)/2]^2 – [(x – 1)/2]^2
= [(x^2 + 2x + 1) / 4] – [(x^2 – 2x + 1) / 4]
= [(x^2 + 2x + 1) – (x^2 – 2x + 1)] / 4
= [x^2 + 2x + 1 – x^2 + 2x – 1)] / 4
= 4x / 4
i.e. y^2 = x â¹ y = √x
This is exactly what we were looking for, the square root of x which is irrational. Now, the length of the third arm of the triangle we constructed can be marked on the Number Line using a compass.
So, if you are looking for the size of √x, this is how we go about it. Mark a point A. Mark B such that AB = x units. Mark C such that BC = 1 unit. That is, AC = x+1. Bisect AC. If D is the point of bisection of AC, AD = DC = (x+1)/2. Now, what would be the length of DB? Since DC = (x+1)/2 and BC = 1, DB = DC – BC = [(x+1)/2] – 1. That is (x+1-2)/2 or (x-1)/2.
Let’s construct the triangle now. Draw a line perpendicular to AC at B. From D, intersect the vertical line at E such that DE = AD. Now we have a right triangle in which the base is (x-1)/2 and the hypotenuse is (x+1)/2. Can you see what the measure of BE is going to be? Of course, as we have shown above, it’s going to be √x. You may transfer this length of BE to the number line now, using the compass.
Try representing √5, √7, √11, √6.8 and √9.5 on the Number Line. Each of them must hardly take a couple of minutes or less.
Math can be fun. As you explore, it is exciting to see how arithmetic, algebra and geometry converge eventually.